Other mistake: The video stream displayed in the night sky showing which tributes have died after the first day does not add up correctly. They display in district order. We see the start, the girl from three (so the boy is still alive), and the boy from four with no breaks. Then they break away to show Katniss and the control room, after which they continuously show the boy and girl from seven, the boy from eight (so the girl is still alive), the girl from nine (so the boy is still alive), and then fade to the end. We know that Foxface, the girl from five, dies later in the movie. Even if we assume that during the portion not shown, the girl from four, the boy from five, and both tributes from six are dead, that's only a total of ten. Thirteen are supposed to be dead at that point in the movie. (01:16:20)
Other mistake: When Katniss retrieves the bow and arrows from Glimmer, there are six arrows in the quiver. She probably recovered the two arrows that Glimmer and Cato shot at her in the tree, bringing the total to eight. Katniss did not recover the two arrows she used to blow up the Careers' supplies, bringing her total down to six. She probably recovered the arrow used to kill Marvel, but did not recover the two arrows she shot at Clove, bringing her total to five. She did not recover the arrow used to kill the muttation, nor did she recover the two arrows used on Cato, meaning that at the end of the Games, she should only have had two arrows remaining. However, at the end, she still had about half a dozen arrows remaining.
Other mistake: When Katniss, Peeta, and Cato are on top of the Cornucopia (after being chased by the muttations), near the end of the film, Katniss aims an arrow at Cato while having Peeta in a headlock. Katniss has her arrow fletchings on backwards. This would prevent her from shooting straight. (02:00:25)
Other mistake: We are told that there are a boy and a girl tribute from each of the 12 districts, making a total of 24 tributes. However, in the scene when Katniss is figuring out the Cornucopia stockpile is mined, the camera sweeps round dug-up earth next to all of the launch pedestals - yet only 20 pedestals can be counted. This can also be seen from an aerial shot in the same scene.
Other mistake: When the children of District 12 are walking to the reaping, just after Prim receives the mockingjay pin, a boy can be seen wearing black and white converse shoes. (00:10:15)